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. The lift to H 3 of a quasi-Fuchsian convex core boundary component.

Introduction
Recent insights into the combinatorial geometry of Teichmüller space have shed new light on fundamental questions in hyperbolic geometry in 2 and 3 dimensions. Paradoxically, a coarse perspective on Teichmüller space appears to refine the analogy of Teichmüller geometry with the internal geometry of hyperbolic 3-manifolds first introduced and pursued by W. Thurston. In this paper we develop such a coarse perspective on the Weil-Petersson metric on Teichmüller space by relating it to a graph of pair-of-pants decompositions of surfaces introduced by Hatcher and Thurston. This viewpoint generates a new connection between the Weil-Petersson geometry of Teichmüller space and the geometry of the convex core of a hyperbolic 3-manifold.
For simplicity, let S be a closed oriented surface of negative Euler characteristic. A pants decomposition of S is a maximal collection of distinct isotopy classes of pairwise disjoint essential simple closed curves on S. We say two distinct pants decompositions P and P ′ are related by an elementary move if P ′ can be obtained from P by replacing a curve α ∈ P by a curve β intersecting α minimally (see Figure 3).
One obtains the pants graph P(S) by making each pants decomposition a vertex and joining two pants decompositions differing by an elementary move by an edge. Setting the length of each edge to 1, P(S) becomes a metric space. We find the graph P(S) provides a combinatorial model for the coarse geometry of the Weil-Petersson metric: Theorem 1.1. The graph P(S) is naturally quasi-isometric to Teichmüller space with the Weil-Petersson metric.
The connection to hyperbolic 3-manifolds is simple to describe. By a theorem of Bers, a pair of points (X, Y ) ∈ Teich(S) × Teich(S) naturally determines a quasi-Fuchsian hyperbolic 3-manifold Q(X, Y ) ∼ = S × R with X and Y in its conformal boundary at infinity. Its convex core, denoted core(Q(X, Y )), is the smallest convex subset of Q(X, Y ) carrying its fundamental group. The convex core is itself homeomorphic to S × I and carries all the essential geometric information about the manifold Q(X, Y ).
Because Q(X, Y ) is obtained from the pair (X, Y ) by an analytic process (Bers's simultaneous uniformization), it is a central challenge in the study of hyperbolic 3manifolds to understand the geometry of Q(X, Y ) purely in terms of the geometry of X and Y . Our main theorem proves a conjecture of Thurston that the following fundamental connection exists between convex core volume and the Weil-Petersson distance.
Theorem 1.2. The volume of the convex core of Q(X, Y ) is comparable to the Weil-Petersson distance d WP (X, Y ).
Here, comparability means that two quantities are equal up to uniform additive and multiplicative error: i.e. there are constants K 1 > 1 and K 2 > 0 depending only on S so that for any (X, Y ) ∈ Teich(S) × Teich(S) we have d WP (X, Y ) K 1 − K 2 ≤ vol(core(Q(X, Y ))) ≤ K 1 d WP (X, Y ) + K 2 .
Throughout the paper we will use the contraction vol(X, Y ) = vol(core(Q(X, Y ))) and the notation ≍ to denote the comparability of two quantities; then Theorem 1.2 becomes d WP (X, Y ) ≍ vol(X, Y ). The volume of the convex core of a complete hyperbolic 3-manifold M = H 3 /Γ is directly related to the lowest eigenvalue of the Laplacian on M as well as the Hausdorff dimension of the limit set Λ(Γ) ⊂ C, namely the complement of the invariant domain of discontinuity Ω(Γ) ⊂ C where the action of Kleinian covering group Γ ⊂ PSL 2 (C) for M is properly discontinuous (see Figure 2 for two examples of limit sets 1 ).
As an immediate application, Theorem 1.2 implies the following new relationship between these analytic invariants and the Weil-Petersson distance. Let λ 0 (X, Y ) denote the lowest eigenvalue of the Laplacian on the quasi-Fuchsian hyperbolic 3manifold Q(X, Y ) = H 3 /Γ(X, Y ) and let D(X, Y ) denote the Hausdorff dimension of the limit set of Γ(X, Y ).  Theorem 1.3. Given S there are constants K > 0, C 1 , C 2 , C 3 , and C 4 > 1 so that if d WP (X, Y ) > K then , Proof: The relation λ 0 (X, Y ) = D(X, Y )(2 − D(X, Y )) follows from a general result by D. Sullivan (see [Sul2,Thm. 2.17]), after applying Bowen's Theorem [Bow] that D(X, Y ) ≥ 1 with equality if and only if X = Y . Theorem 1.2 may be rephrased to claim the existence of K, K ′ so that for d WP (X, Y ) > K we have The theorem then follows from the double inequality c 1 vol(X, Y ) 2 ≤ λ 0 (X, Y ) ≤ c 2 vol(X, Y ) (see [BC, Main Thm.] and [Can1, Thm. A]) after collecting constants.
The pants graph. Since Theorem 1.2 relies directly on Theorem 1.1 we detail our coarse perspective on the Weil-Petersson metric.
To describe the nature of the quasi-isometry between the graph P(S) and the Weil-Petersson metric, we recall that by a theorem of Bers, there is a constant L > 0 depending only on S so that for each X ∈ Teich(S) there is a pants decomposition P so that ℓ X (α) < L for each α ∈ P, A compactness argument shows that each X ∈ V (P ) = V L (P ) lies a uniformly definite distance from ∂V 2L (P ). Thus, a unit-length Weil-Petersson geodesic can always be covered by a uniform number of sub level sets V 2L (P ). It follows that any pair of pants decompositions P and P ′ for which V (P ) and V (P ′ ) contain the endpoints of a unit length Weil-Petersson geodesic, P and P ′ have uniformly bounded distance in P(S), and the theorem follows.
Outline of the proof of Theorem 1.2. The proof has two parts.
Bounding volume from below: The bound below of core volume in terms of the Weil-Petersson distance begins with an interpolation through the convex core h t : Z t → core(Q(X, Y )) of 1-Lipschitz maps of hyperbolic surfaces. It follows that for each essential simple closed curve α we have inf The path Z t , then, only passes through sets V (P ) for which each element in P has length less than L in Q(X, Y ). Applying recent work of Masur and Minsky, we show if a sequence {P 1 , . . . , P n } of pants decompositions is built from N curves and makes bounded jumps, i.e. d P (P j , P j+1 ) < k, then its ends satisfy the bound d P (P 1 , P n ) < K 0 N , where K 0 depends only on k and S.
The Margulis lemma forces closed geodesics with length less than L in Q(X, Y ) that represent different isotopy classes to be uniformly equidistributed through the convex core. Since each such representative makes a definite contribution to core volume, the lower bound follows.
Bounding volume from above. Given pants decompositions P X and P Y so that X ∈ V (P X ) and Y ∈ V (P Y ), and a geodesic G ⊂ P(S) joining P X to P Y , we consider the closed geodesics spin(G) = {α * | α ∈ P for P ∈ G} where α * denotes the geodesic representative of α in Q(X, Y ). We build a straight triangulation T of all but a uniformly bounded volume portion of core(Q(X, Y )) so that vertices of T lie on α * ∈ spin(G), the so-called spinning geodesics.
Our triangulation has the property that all but constant times d P (P X , P Y ) of the tetrahedra in T have at least one edge in a spinning geodesic α * . We then use a spinning trick: by homotoping the vertices around the geodesics in spin(G) keeping the triangulation straight, all tetrahedra with an edge in any α * can be made to have arbitrarily small volume.
Since there is an a priori bound to the volume of a tetrahedron in H 3 , the remaining tetrahedra have uniformly bounded volume. The theorem then follows from the comparability d WP (X, Y ) ≍ d P (P X , P Y ).
Geometrically finite hyperbolic 3-manifolds. We remark that simple generalizations of these techniques may be employed to obtain estimates for core volume of hyperbolic 3-manifolds that are not quasi-Fuchsian once the appropriate version of Weil-Petersson distance is defined. For example, given a hyperbolic 3-manifold M ψ that fibers over the circle with monodromy ψ, the volume of M ψ is comparable to the Weil-Petersson translation distance of ψ (with constants depending only on the topology of the fiber). We take up these generalizations in [Br3].
Algebraic and geometric limits. As an application of Theorem 1.2, boundedness of the Weil-Petersson distance d WP (X k , Y k ) for sequences predicts the geometric finiteness of the geometric limit of Q(X k , Y k ).
The space QF (S) of all quasi-Fuchsian hyperbolic 3-manifolds lies in the space AH(S) of all complete hyperbolic 3-manifolds M marked by homotopy equivalences (h : S → M ) so that h * sends peripheral elements of π 1 (S) to parabolic elements of π 1 (M ). The space AH(S) carries the algebraic topology or the compact-open topology on the induced representations h * : π 1 (S) → Isom + (H 3 ) up to conjugacy.
In an algebraically convergent sequence {(h k : S → M k )} in AH(S), normalizing the induced representations ρ k = (h k ) * to converge on generators one may always extract a subsequence so that the groups ρ k (π 1 (S)) = Γ k converge in the Gromov-Hausdorff topology on discrete subgroups of Isom + (H 3 ), or geometrically, to a limit Γ G . A central issue in the deformation theory of hyperbolic 3-manifolds is to understand the geometric limit Applying Theorem 1.2, we obtain the following criterion: Note that geometric finiteness of N G implies geometric finiteness of Q ∞ but not conversely.
History and references. The fundamental properties of the Weil-Petersson metric we use are discussed in [Wol1], [Wol3], [Wol4] and [Mas]. The pants graph is the 1-skeleton of the pants complex, introduced in [HT] (see also [HLS]) which is there proven to be connected. The relation of the pants graph to the Weil-Petersson metric is similar in spirit to the relative hyperbolicity theorem for Teichmüller space of [MM1] where the (related) complex of curves is shown to be quasi-isometric to the electric Teichmüller space, and to be Gromov-hyperbolic (the pants complex and the Weil-Petersson metric are not in general Gromov-hyperbolic [BF]). For more on quasi-Fuchsian manifolds and their algebraic and geometric limits, see [Th1], [Brs1], [Mc2], [Mc1], [Br2], and [Ot].
Plan of the paper. After discussing the fundamental work of S. Wolpert and H. Masur on the Weil-Petersson metric that will serve as our jumping off point in section 2, we prove the comparability of Weil-Petersson distance and pants distance (Theorem 1.1) in section 3. We then establish the lower bound on vol(X, Y ) in terms of the distance d P (P X , P Y ) in section 4. Section 5 applies the combinatorics of pants decompositions along a geodesic G ⊂ P(S) joining P X to P Y to bound volume from above in terms of pants distance. Theorem 1.2 then follows from the comparability of Theorem 1.1. We conclude with applications to the study of geometric limits, proving Theorem 1.4. and Curt McMullen for useful conversations, and to Lewis Bowen and the referee for corrections and suggestions.

The extended Weil-Petersson metric
Let S be a compact oriented surface of negative Euler characteristic. We allow S to have boundary and let int(S) denote its interior. Let S denote the set of isotopy classes of essential, non-peripheral, simple closed curves on S.
A pants decomposition P ⊂ S is a maximal collection of isotopy classes with pairwise disjoint representatives on S. The usual geometric intersection number i(α, β) of a pair (α, β) ∈ S × S generalizes to a total intersection number i(P, P ′ ) of pants decompositions by summing the geometric intersections of their components.
The Teichmüller space Teich(S) of S parameterizes finite area hyperbolic structures on S up to isotopy. Points in Teich(S) are pairs (f, X) where X is a finite area hyperbolic surface X equipped with a homeomorphism f : int(S) → X, up to the equivalence (f, for each X ∈ Teich(S), indicating X is assembled from hyperbolic pairs of pants with boundary lengths prescribed by ℓ X (α i ) glued together twisted by θ X (α i ). (For more on Teichmüller space and Fenchel-Nielsen coordinates see [IT] or [Gard]).
The Weil-Petersson metric. Each X ∈ Teich(S) is naturally a complex 1manifold via its uniformization X = H 2 /Γ as the quotient of the upper half plane by a Fuchsian group. The Teichmüller space has a complex manifold structure of dimension 3g − 3 + n where S has genus g and n boundary components.
The space of holomorphic quadratic differentials Q(X) on X ∈ Teich(S) (holomorphic forms of type φ(z)dz 2 on X) is naturally the cotangent space T * X Teich(S) to Teich(S) at X. The Weil-Petersson metric on Teich(S) unifies the hyperbolic and holomorphic perspectives on X: it arises from the L 2 inner product on Q(X), namely ϕ, ψ WP = X ϕψ ρ 2 where ρ(z)|dz| is the hyperbolic metric on X, by the usual pairing (µ, ϕ) X = X µϕ between T X Teich(S) and T * X Teich(S) (see, e.g. [Wol3,Sec. 1]). In what follows, we will be interested only in the Riemannian part g WP of the Weil-Petersson metric, and its associated distance function d WP (., .) on Teich(S).
The Weil-Petersson metric has negative sectional curvature [Tro] [Wol2], and the modular group Mod(S) (the group of isotopy classes of orientation preserving homeomorphisms of S) acts by isometries of g WP . Thus, g WP descends to a metric on the Moduli space M(S) = Teich(S)/Mod(S).
Work of S. Wolpert shows two important properties of the Weil-Petersson metric we will use: WPI The Weil-Petersson metric is not complete: "pinching geodesics" in the Teichmüller metric (which leave every compact set of Teich(S)) have finite Weil-Petersson length [Wol1].
The augmented Teichmüller space. In [Mas], H. Masur shows the Weil-Petersson metric extends to the augmented Teichmüller space Teich(S) obtained by adding boundary Teichmüller spaces consisting of marked noded Riemann surfaces, which we now describe (see [Brs3] for a detailed discussion).
A Riemann surface with nodes W is a connected complex space so that each point p ∈ W has a neighborhood isomorphic to {z ∈ C | |z| < 1} or isomorphic to {(z, w) ∈ C 2 | |z| < 1, |w| < 1, and zw = 0} by an isomorphism sending p to (0, 0) ∈ C 2 . In the latter case, p is called a node of X. The complement of the nodes is a union of Riemann surfaces called the pieces of W . We say W is hyperbolic if each piece of W admits a complete finite-area hyperbolic structure.
Given curves ν 1 , . . . , ν j in a pants decomposition P = {α 1 , . . . , α |P | } of S, a marked noded hyperbolic surface pinched along ν 1 , . . . , ν j is a noded hyperbolic Riemann surface W together with a continuous map in the product Teichmüller space by taking the restriction of f to each component of int(S) − ν 1 ∪ . . . ∪ ν j as a marking on each piece of W . A marked piece of W l ∈ Teich(S l ), 1 ≤ l ≤ k, has Fenchel-Nielsen coordinates with respect to the elements of the pants decomposition P that lie in S l .
Two marked hyperbolic noded surfaces (f 1 , W 1 ) and (f 2 , W 2 ) are equivalent, if there is continuous map φ : W 1 → W 2 that is isometric on each piece of W 1 for which φ • f 1 = f 2 after precomposition with an isotopy of S.
The augmented Teichmüller space Teich(S) is obtained by adjoining equivalence classes of marked noded hyperbolic surfaces to Teich(S). The topology on Teich(S) is given as follows. Given a pants decomposition P , and a point W ∈ Teich(S) with curves ν 1 ∪ . . . ∪ ν j in P pinched to nodes, we extend the Fenchel-Nielsen coordinates to W by defining the coordinates ℓ W (ν i ) = 0. Then a neighborhood of W in Teich(S) consists of (possibly noded) hyperbolic Riemann surfaces X whose length coordinates ℓ X (α p ) are close to those of W for p = 1, . . . , 3g − 3 + n, and whose twist coordinates θ X (α p ) are close to those of X for each p such that α p = ν i . (see [IT, App. B]).
The Weil-Petersson metric extends to the augmented Teichmüller space as its completion (see [Mas]), giving a Mod(S) invariant metric on Teich(S). The quotient Teich(S)/Mod(S) = M(S), the familiar Mayer-Mumford-Deligne compactification of the moduli space (see [Brs3]), inherits a complete extension of the Weil-Petersson metric on M(S). We denote the corresponding distance by Evidently, the failure of completeness of the Weil-Petersson metric occurs at limits of pinching sequences X t for which the length coordinates ℓ Xt (ν i ) tend to zero for some collection of curves in a pants decomposition P .
Given a pants decomposition P and a collection α 1 , . . . , α k of curves in P , the minimal distance from a point X ∈ Teich(S) to a noded Riemann surface Z with nodes along α 1 , . . . , α k is estimated in terms of the geodesic length sum of the lengths of α i on X by Remark: This estimate is a recent improvement of similar estimates originally obtained in [Wol4,Ex. 4.3] and cited in earlier versions of this manuscript.
Sub level sets. We recall the following theorem of Bers.
Theorem 2.1 (Bers). There is a constant L > 0 depending only on S such that for any X ∈ Teich(S) there is a pants decomposition P such that ℓ X (α) < L for each α ∈ P .
We call this L the Bers constant for S. Given a pants decomposition P , and a positive real number ℓ ∈ R + , we consider the sub level set Then by Bers's theorem, the union of the sets V L (P ) over all pants decompositions gives an open cover of Teich(S). Because L depends only on S, we abbreviate V (P ) = V L (P ).
Then we have the following: Proposition 2.2. The sub level sets V (P ) have the following properties.
(1) Each V (P ) is convex in the Weil-Petersson metric, and (2) there is a constant D > 0, depending only on S, for which the Weil-Petersson diameter diam WP (V (P )) < D.
Proof: Geodesic convexity of V (P ) follows immediately from WPII, the convexity of the geodesic length functions ℓ X (.)(α) for each α ∈ P . To see each V (P ) has bounded Weil-Petersson diameter, let W P be the (unique) maximally noded Riemann surface where each curve in P is pinched. By equation (2.1) there is a constant C(L) so that for each X ∈ V (P ) we have d WP (X, W P ) < C(L).
By the triangle inequality for d WP , if X and Y lie in V (P ) then the distance d WP (X, Y ) is bounded by 2C(L). By geodesic convexity of V (P ) the geodesic joining X to Y lies in V (P ), so we have the bound on d WP (X, Y ) which we set equal to D.

A combinatorial Weil-Petersson distance
In this section, we relate the coarse geometry of the Weil-Petersson metric to the pants graph P(S) defined in the introduction. We do this by exhibiting a quasi-isometry between the two spaces with their respective distances.
Definition 3.1. Given k 1 > 1 and k 2 > 0, a map f : (X, d) → (Y, d ′ ) of metric spaces is a (k 1 , k 2 )-quasi-isometric embedding if for each pair of points x and y in The spaces (X, d) and (Y, d ′ ) are quasi-isometric if for some k 1 > 1 and k 2 > 0 there are (k 1 , k 2 )-quasi-isometric embeddings from (X, d) to (Y, d ′ ) and from (Y, d ′ ) to (X, d). In practice, it suffices to exhibit a quasi-isometry from (X, d) to (Y, d ′ ), namely, a quasi-isometric embedding with uniformly dense image. Given such a quasi-isometry from (X, d) to (Y, d ′ ), a quasi-isometric embedding from (Y, d ′ ) to (X, d) is readily constructed, so the spaces are quasi-isometric.

Let
Q : P 0 (S) → Teich(S) be any embedding of the vertices P 0 (S) of P(S) into Teich(S) so that Q(P ) lies in V (P ). The main theorem of this section is the following: Theorem 3.2. The map Q is a quasi-isometry of the 0-skeleton P 0 (S) of P(S) with Teich(S) with its Weil-Petersson distance.
Proof: By the uniform bound diam WP (V (P )) < D on the diameter of V (P ), the image Q(P(S)) is D-dense in Teich(S). It suffices, then, to show that there are uniform constants A 1 ≥ 1 and A 2 ≥ 0 so that 1 We first show that the map Q is 2D-Lipschitz. Given P 0 and P 1 such that d P (P 0 , P 1 ) = 1, P 0 and P 1 differ by a single elementary move. Let α ∈ P 0 and β ∈ P 1 be the curves involved in this elementary move, i.e. P 0 − α = P 1 − β and i(α, β) = 1 or 2 depending on whether the component S α ⊂ S − (P 0 − α) containing α is a punctured torus or four-times punctured sphere. Let Z ∈ Teich(S α ) be the "square" punctured torus: i.e. Z is obtained by identifying opposite sides of an ideal square in H 2 with order-4 rotational symmetry about the origin in the disk model of H 2 . Marking Z so that the common perpendiculars to the opposite sides descend to closed geodesics α and β on Z (see Figure 4), we have by symmetry that the length ℓ Z (α) equals the length ℓ Z (β), which is the shortest length of any non-peripheral simple closed curve on Z. In particular, we have Let (f, W ) ∈ Teich(S) be the noded Riemann surface with nodes at each γ ∈ P −α, and one non-rigid piece f | Sα → Z. For any Z ′ ∈ Teich(S) sufficiently close to α β Z Z, if Z ′ ∈ V (P ′ ) then P ′ contains an essential non-peripheral curve in S α . Taking Z ′ arbitrarily close to Z, we may conclude that L(S α ) ≤ L(S) = L. Then for any Riemann surface Z ′ ∈ Teich(S) sufficiently close to W ∈ Teich(S) we have ℓ Z ′ (γ) < L for each γ ∈ P 0 ∪ P 1 . In other words, Z ′ lies in the intersection V (P 0 ) ∩ V (P 1 ). Letting Z be the double of the symmetric ideal square described above, a similar argument handles the genus-0 case.
Therefore we may conclude that d WP (Q(P 0 ), Q(P 1 )) < 2D when d P (P 0 , P 1 ) = 1, so by the triangle inequality, Q is 2D-Lipschitz. To show that for some A 1 and A 2 the inequality 1 holds is somewhat more delicate. We break this into a series of lemmas.
Lemma 3.3. Given L ′ > L, there is an integer B > 0 so that given P and P ′ in Proof: The hypotheses imply that there is some X ∈ Teich(S) so that ℓ X (α) < L ′ for each α ∈ P ∪ P ′ . By an application of the collar lemma [Bus,Thm. 4.4.6] there is a constant C depending only L ′ and S so that the total geometric intersection number i(P, P ′ ) satisfies i(P, P ′ ) ≤ C. Let Tw(P ) ∼ = Z |P | denote the subgroup of Mod(S) generated by Dehn twists about the curves in P . Then the function i(P, .) : P 0 (S) → Z descends to a function i(P, .) : P 0 (S)/Tw(P ) → Z whose sub level sets are bounded: in other words there are only finitely many equivalence classes Since d P (P, .) also descends to a function is a unit-length Weil-Petersson geodesic joining X 0 and X 1 , then there exist pants We claim that there is an ǫ 0 depending only on L, and L ′ so that The function d L ′ ,P (X) naturally extends to the metric completion ∂V (P ) of ∂V (P ), and d L ′ ,P (X) is invariant under the action of Tw(P ).
Fenchel-Nielsen coordinates for Teich(S) adapted to the pants decomposition P . To extend these Fenchel Nielsen coordinates to the completion, we denote by where each point with ℓ j = 0 for some j lies in the completion. The extended isometric action of Tw(P ) on Teich(S) is cocompact on V L ′ (P ), since Tw(P ) preserves each length coordinate and acts by translations on each twist coordinate.

THE WEIL-PETERSSON METRIC AND VOLUMES OF CONVEX CORES 13
In these extended Fenchel-Nielsen coordinates, the completion ∂V L ′ (P ) of the boundary ∂V L ′ (P ) is the locus of coordinates for which ℓ j = L ′ for some j ∈ 1, . . . , |P |. Thus is a closed subset of the compact set V L ′ (P )/Tw(P ) and is thus compact.
Since the quotients ∂V L ′ (P )/Tw(P ) and ∂V (P )/Tw(P ) are disjoint compact subsets of V L ′ (P )/Tw(P ), it follows that the function . It follows that after setting J equal to the least integer greater than 2/ǫ 0 , we may select from the pants decompositions P 1 , . . . , P m pants decompositions P 1 , . . . , P J (possibly with repetition) so that To complete the proof of Theorem 3.2, let X t be the Weil-Petersson geodesic joining arbitrary distinct Riemann surfaces X and Y in Teich(S). Let P X and P Y be pants decompositions for which X lies in V (P X ) and Y lies in V (P Y ). Let I(P ) ⊂ [0, 1] denote the values of t for which X t ∈ V 2L (P ). By convexity of V (P ) (Proposition 2.2) each I(P ) is an interval.
Taking L ′ = 2L, Lemma 3.4 provides a J > 0 and a sequence {P j } N j=0 ∈ P(S) so that • X(t) is covered by the union ∪ j I(P j ), • the least upper bound of I(P j ) lies in I(P j+1 ), and Moreover, for successive pants decompositions P j , P j+1 , we have Combining equations 3.2 and 3.3, we have where B and J depend only on L which depends only on S. Setting A 1 = BJ and A 2 = 1 concludes the proof of Theorem 3.2.
4. Bounding the core volume from below To simplify notation, let and recall that vol(X, Y ) denotes the convex core volume vol(core(Q(X, Y ))). In this section we prove Theorem 4.1. Given S, there are constants K 1 > 1 and K 2 > 0 so that The proof is given as a series of lemmas. Fix attention on a given quasi-Fuchsian manifold Q(X, Y ). Given a constant L 0 > 0, let S <L0 ⊂ S denote the set of isotopy classes Theorem 4.1 will follow from a linear lower bound on vol(X, Y ) given in terms of the size of S <L (Lemma 4.8) and the following lemma.
Lemma 4.2. Let P X and P Y be pants decompositions so that X ∈ V (P X ) and Y ∈ V (P Y ). Then there is a constant K depending only on S so that The lemma will follow from the following general result on paths in P(S) that are built out of a given collection of curves in S. For reference, let π S : P 0 (S) → S denote the projection that assigns to each P the collection of curves used to build it.
Lemma 4.3. Let k ∈ N and let g = {P I = P 0 , . . . , P N = P T } ⊂ P(S) be a sequence of pants decompositions with the property that d P (P j , P j+1 ) < k. Let S g ⊂ S denote the image of g under the projection π S . There is a constant K 0 > 0 depending on k and S so that The complex of curves. The graph P(S) is related to the complex of curves C(S), introduced by W. Harvey [Har]. To prove Lemma 4.3 we describe recent work of Masur and Minksy on C(S). The main result of [MM1] shows that C(S) is in fact a Gromov hyperbolic metric space with the metric obtained by making each simplex a standard Euclidean simplex. Its sequel [MM2] introduces a theory of hierarchies of so-called "tight geodesics" in C(S) and in sub-complexes C(Y ) for essential subsurfaces Y ⊂ S. Such hierarchies and their hyperbolicity properties play an integral role in our control of volume.
To describe the topological type of S, we let where S has genus g with n boundary components. We consider only those surfaces S for which int(S) admits a hyperbolic structure (so d(S) > 0). The complex of curves C(S) is a simplicial complex with 0-skeleton S, and higher dimensional simplices described as follows: • for d(S) > 1 and k ≥ 1, k-simplices of C(S) span k + 1-tuples α 1 , . . . , α k+1 of vertices for which i(α i , α j ) = 0, and • if d(S) = 1, C(S) is a 1-complex whose edges join vertices α and α ′ in C(S) that intersect minimally; i.e. C(S) = P(S). Given an essential subsurface Y ⊂ S with d(Y ) ≥ 2, the curve complex C(Y ) is naturally a subcomplex of C(S). Given a set W let P(W ) denote its power set, i.e. the set of all subsets of W . Masur and Minsky define a projection in C(Y ) of A and B. Note that while d Y (., .) is more a diameter than a distance when A and B are close, it gives a useful notion of distance between sets of bounded diameter and does satisfy the triangle inequality.
By [MM2,Lem. 2.3] the projection π Y has a Lipschitz property: if ∆ is a simplex in C(S) so that ∆ intersects Y , then we have diam C(Y ) (π Y (∆)) ≤ 2. If P I and P T are two subsets of C(S), letting π Y (P I ) = ∪ α∈PI π Y (α), and likewise for P T , then the projection distance d Y (P I , P T ) between P I and P In particular, if P and P ′ are pants decompositions that differ by a single elementary move, then we have d Y (P, A central theorem we will use is the following: Theorem 4.4 (Thm. 6.12 of [MM2]). There is a constant M 0 (S) so that given M > M 0 there exist c 0 and c 1 so that if P I and P T are pants decompositions in P(S) then we have where the sum is taken over all non-annular essential subsurfaces Y ⊆ S satisfying We apply this result to prove Lemma 4.3. Our argument is quite similar to that of [MM2,Thm. 6.10], where it is shown that a given pants decomposition along an elementary move sequence can contribute to progress in only boundedly many projections to subsurfaces simultaneously. We seek the analogous statement for a single curve occuring in pants decompositions joining P I to P T .
Proof: (of Lemma 4.3). To prove the lemma, we will relate the sum of the projections to the size of S g . To do this, we note that when the projection distance d Y (P I , P T ) is large, there must be a definite portion of the projection of g to Y that is far from both π Y (P I ) and π Y (P T ) in d Y (., .); this follows from the triangle inequality for d Y and the fact that elementary moves in P(S) make Lipschitz progress as measured by d Y .
We argue that a given curve α can contribute only to a bounded amount of progress in boundedly many different subsurfaces. Precisely, let Y and Z be two essential, intersecting, non-annular subsurfaces of S, neither of which is contained in the other. A lemma of Masur and Minsky [MM2,Lem. 6.11] enforces a partial ordering "≺" on such subsurfaces with respect to the pants decompositions P and P ′ , provided the projection distances d Y (P, P ′ ) and d Z (P, P ′ ) are greater than a constant M 2 depending only on S. Taking M 2 to be the constant of [MM2, Lem. 6.2] with the same name, we say the subsurfaces Y and Z are (P, We rephrase [MM2, Lem. 6.11] as follows. where Y ≺ Z represents the case h ≺ t k and Z ≺ Y represents the case k ≺ t h.
Given a subset A ⊂ [1, N ], we denote by the diameter of the projection of the pants decompositions with indices in A to the curve complex C(Y ).
Given α ∈ S g for which π Y (α) = ∅, we denote by J Y (α) ⊂ J Y the subset for which if i ∈ J Y (α) then α lies in P i . We make three observations for later reference: Let Y ⊆ S and Z ⊆ S be two non-annular intersecting subsurfaces neither of which is contained in the other so that each contributes to the sum of Theorem 4.4: i.e. we have d Y (P I , P T ) > M and d Z (P I , P T ) > M. This assumption guarantees, in particular, that Y and Z are (P I , P T )-ordered. We make the following claim: ( * ) If J Y (α) is non-empty, then J Z (α) must be empty. Arguing by contradiction, assume , since α lies in P i and in P j . The same conclusion holds with Z in place of Y .
Since i lies in J Y (α) we have d Y (P I , P i ) > M 4 and d Y (P i , P T ) > M 4 , so it follows that d Y (P I , P j ) ≥ M 4 − 4 and d Y (P j , P T ) ≥ M 4 − 4. As j lies in J Z (α) we have d Z (P I , P j ) ≥ M 4 , and since M 4 − 4 > M 2 , it follows that Y and Z are also (P I , P j )-ordered. Let ≺ j denote the (P I , P j )-ordering and assume without loss of generality that Y ≺ j Z. Then applying Lemma 4.5 we have Since Y and Z are also (P I , P T )-ordered, we may first assume that Y ≺ Z. Then Lemma 4.5 gives d Y (∂Z, P T ) < M 3 which implies that contradicting the assumption that j ∈ J Z (α). If on the other hand we have Z ≺ Y , then Lemma 4.5 gives d Y (P I , ∂Z) < M 3 from which we conclude d Y (P I , P j ) < 2M 3 , which contradicts the same assumption. Thus, either J Y (α) or J Z (α) must be empty, and the claim ( * ) is proven.
Applying observations (I) and There is a uniform bound s depending only on S to the size of any collection of subsurfaces any pair of which is disjoint or nested (see [MM2,Lem. 6.10, proof]) so by our claim ( * ), the number of Y for which J Y (α) can be non-empty is bounded by s. Thus we have Applying observation (III), we have d Y (P j , P j+1 ) < 4k. Thus, for each Y satisfying d Y (P I , P T ) > M the set J Y is in particular non-empty, and we have Since M = max{M 0 , 4M 4 }, applying Theorem 4.4 there are constants c 0 and c 1 so that we have Since |S g | is always at least d(S), we may combine all of the above constants into a single K 0 for which d P (P I , P T ) ≤ K 0 |S g |.
To prove Lemma 4.2, we will apply Lemma 4.3 to a sequence of pants decompositions {P X = P 0 , . . . , P N = P Y } so that π S (P j ) ⊂ S <L for each j, and so that d P (P j , P j+1 ) is bounded by an a priori constant. The existence of such a sequence is provided by an interpolation of 1-Lipschitz homotopy equivalences of hyperbolic surfaces into Q(X, Y ) that pass from one side of the convex core to the other. The existence of such an interpolation follows from work of R. Canary on simplicial hyperbolic surfaces which we now describe.
Simplicial hyperbolic surfaces. Let Sing k (S) denote the finite-area marked singular hyperbolic structures on S: complete finite area hyperbolic surfaces Z with at most k cone singularities, each with cone-angle at least 2π, equipped with marking homeomorphisms h : int(S) → Z up to marking-preserving isometry. Roughly speaking, a simplicial hyperbolic surface is a path-isometric mapping from a singular hyperbolic surface to a hyperbolic 3-manifold that is totally geodesic in the complement of a "triangulation." We now make this notion precise.
If V contains a point in each boundary component of the compact surface S, then a triangulation of S is a maximal curve system in A(S, V ). Likewise, we may view the interior int(S) of S as a "punctured surface" by collapsing each boundary component γ ⊂ ∂S to a point v γ to obtain a surface R. Then we have If V is a subset of R containing ∪ γ v γ , then a triangulation of int(S) is the restriction of a maximal curve system in A(R, V ) restricted to R − {v γ | γ ⊂ ∂S} = int(S). Note that in each definition, an edge may have its boundary vertices identified and a face may have boundary edges identified.
The main result of [Hat] guarantees that any two triangulations in A(S, V ) are related by a finite sequence of elementary moves (see Figure 5). Let T be a triangulation of int(S) in the above sense, and let h : int(S) → Z be a singular hyperbolic surface for which h is isotopic to a map with the property that each cone singularity of Z lies in the image of a vertex of T . Isotope h to send each edge of T to its geodesic representative rel-endpoints on Z (if an edge e terminates at a puncture, h should send e to a geodesic arc asymptotic to the corresponding cusp of Z). Then if N is a hyperbolic 3-manifold and there is a path-isometry g : Z → N that is a local isometry on Z − T , then we call the pair (g, Z) a simplicial hyperbolic surface in N with associated triangulation T .
Often the construction goes in the other direction: given a triangulation T of int(S) with one vertex v ∈ int(S) and at least one edge e so that e ∪v forms a closed loop γ, we can straighten any smooth, proper, incompressible map g ′ of int(S) to N to a simplicial hyperbolic surface with associated triangulation T . First we straighten g ′ so it maps γ to its geodesic representative. Then, straightening g ′ on the edges of T rel-endpoints (possibly ideal endpoints) and then on faces of T , we obtain a map g : int(S) → N . The pull-back metric from N determines a singular hyperbolic surface Z with a cone singularity at v; since v lies on a closed geodesic that is mapped to a closed geodesic in N , the cone angle at v is at least 2π. The result is a simplicial hyperbolic surface (g, Z) in N with associated triangulation T . In this case we say that (g, Z) is adapted to γ.
Let (f : S → N ) ∈ AH(S). Then we denote by SH k (N ) the marking preserving simplicial hyperbolic surfaces in N with at most k cone-singularities, namely, simplicial hyperbolic surfaces (g, Z), with (h : int(S) → Z) ∈ Sing k (S), so that g • h is homotopic to f . If σ is a simplex in C(S) with vertices α 1 , . . . , α p , then we say a simplicial hyperbolic surface (g, Z) ∈ SH k (N ) realizes σ if g maps each α i isometrically to its geodesic representative in N .
Recall that a manifold (f : S → N ) ∈ AH(S) has an accidental parabolic if there is a non-peripheral element γ ∈ π 1 (S) so that f * (γ) is a parabolic element of Isom + (H 3 ). Applying Hatcher's theorem [Hat] allowing one to connect triangulations by elementary moves, Canary proves the following (see [ (Canary). Let N ∈ AH(S) have no accidental parabolics, and let (g 1 , Z 1 ) and (g 2 , Z 2 ) lie in SH 1 (N ) where (g 1 , Z 1 ) is adapted to α and (g 2 , Z 2 ) is adapted to β. Then there is a continuous family (g t : Z t → N ) ⊂ SH 2 (N ), t ∈ [1, 2].
Using such an interpolation, we now give the proof of Lemma 4.2.
Proof: (of Lemma 4.2). Given the quasi-Fuchsian manifold Q(X, Y ), let P X and P Y denote pants decompositions for which X lies in V (P X ) and Y lies in V (P Y ).
Let Z X and Z Y denote simplicial hyperbolic surfaces realizing P X and P Y in Q(X, Y ) and let T X and T Y denote their associated triangulations. Let Z h X ∈ Teich(S) be the hyperbolic surface conformally equivalent to Z X , and let Z h Y be the hyperbolic surface conformally equivalent to Z Y . Finally, let P I and P T be pants decompositions so that Z h X ∈ V (P I ) and Z h Y ∈ V (P T ). The next step will be to interpolate simplicial hyperbolic surfaces between Z X and Z Y and estimate the minimum number of sets V (P j ) the corresponding conformally equivalent hyperbolic representatives in Teich(S) intersect. To show we have not sacrificed too much distance in the pants graph, we prove the following: for each γ ∈ P X . By a lemma of Ahlfors [Ah] we have for each α ∈ P X ∪ P I . Since curves in P X ∪ P I are realized with bounded length on the simplicial hyperbolic surface Z X ∈ SH k (Q(X, Y )), we may apply [Br1, Lem. 3.3] to find a C ′ > 0 depending only on L so that i(P X , P I ) < C ′ .
Arguing similarly for Z Y , we have i(P Y , P T ) < C ′ .
Applying the proof of Theorem 3.3, we have a B ′ > 0 depending only on C ′ for which max{d P (P X , P I ), d P (P Y , P T )} < B ′ .
To complete the proof of Lemma 4.2, we seek a continuous family of simplicial hyperbolic surfaces in Q(X, Y ) interpolating between (h X , Z X ) and (h Y , Z Y ). We first connect (h X , Z X ) and (h Y , Z Y ) to simplicial hyperbolic surfaces (h ′ X , Z ′ X ) and (h ′ Y , Z ′ Y ) adapted to single curves α ∈ P X and β ∈ P Y by continuous families, and then apply the interpolation arguments of Canary. This is easily done by collapsing edges of T X that join distinct vertices of T X down to a single vertex: if e is such an edge adjacent to a vertex v on α, then we may effect such a collapsing by 'dragging h X along h X (e)' (see [Can3,§5]). Precisely, if v ′ is the other vertex in ∂e, we construct a homotopy of (h X , Z X ) to a new simplicial hyperbolic surface by pulling the image h X (v ′ ) along the geodesic segment h X (e) to h X (v) and pulling the edges and faces adjacent to v along with it while keeping the triangulation straight: the image of each triangle is required to lift to the convex hull of its vertices throughout the homotopy.
It is easy to check that under such a collapsing the cone angles at the vertices remain at least 2π, and the number of vertices in T is reduced by 1. We successively collapse edges joining v to different vertices until we are left with a triangulation with the single vertex v, and a simplicial hyperbolic surface (h ′ X , Z ′ X ) ∈ SH 1 (Q(X, Y )) adapted to α.
We perform analogous collapsings on the associated triangulation for (h Y , Z Y ) to obtain the simplicial hyperbolic surface (h ′ Y , Z ′ Y ) ∈ SH 1 (Q(X, Y )) adapted to β. By Theorem 4.6, we may interpolate between (h ′ X , Z ′ X ) and (h ′ Y , Z ′ Y ) by a continuous family of simplicial hyperbolic surfaces, so we have the desired continuous family (h t : Z t → C(X, Y )) ⊂ SH k (Q(X, Y )), t ∈ [0, 1] so that (h 0 , Z 0 ) = (h X , Z X ) and (h 1 , Z 1 ) = (h Y , Z Y ). The singular hyperbolic structures Z t determine a continuous path (h h t : int(S) → Z h t ) ⊂ Teich(S), where as before Z h t is the finite-area hyperbolic structure on S in the same conformal class as Z t . Since Z t and Z h t represent metrics on the same underlying surface int(S), we have a natural continuous family of 1-Lipschitz mappingŝ h t : Z h t → C(X, Y ) of hyperbolic surfaces Z h t ∈ Teich(S) into Q(X, Y ) so that for each t,ĥ t factors through the simplicial hyperbolic surface h t : Z t → C(X, Y ).
There are pants decompositions P 1 , . . . , P N (possibly with repetition) that determine an open cover {U j } N j=0 of [0, 1] so that if t ∈ U j then Z h t lies in V (P j ), and so that U j ∩ U j+1 = ∅ for each j = 0, . . . , N − 1. Applying Lemma 3.3, the sequence of pants decompositions g = {P I = P 0 , . . . , P N = P T } satisfies the hypotheses of Lemma 4.3 with k = B. Applying Lemma 4.3, we have a K 0 so that d P (P I , P T ) ≤ K 0 |S g |.
Since the non-empty set S g is a subset of S <L , and Lemma 4.7 guarantees we may combine constants to obtain a K for which Given a hyperbolic 3-manifold M , we let G <L (M ) denote the set of homotopy classes of closed geodesics in M with length bounded above by a constant L > 0. We use the contraction G <L = G <L (M ) when the manifold M is understood. The next lemma shows that the size |G <L | of G <L provides a lower bound for the convex core volume of a hyperbolic 3-manifold in a general context. Each isotopy class β ∈ G <L has a representative β ⋆ ⊂ core(M ) ≥ǫ with arclength less than L. Since the ǫ/2-balls about points in V cover core(M ) ≥ǫ , each β ⋆ intersects B(x, ǫ/2) for some x ∈ V. Given x ∈ V, let A x ⊂ G <L denote the set Lifting to the universal cover so that x lifts to the origin 0 ∈ H 3 , the elements β ∈ A x determine pairwise disjoint translates of the ball B(0, ǫ/2) ⊂ H 3 lying within the ball B(0, L + 2ǫ) ∈ H 3 . It follows that the number of elements in each A x satisfies |A x | < vol(B(0, L + 2ǫ)) vol(B(0, ǫ/2)) which we set equal to C 0 .
Proof: (of Theorem 4.1). Since S <L is a subset of G <L (Q(X, Y )), we may combine Lemma 4.2 with Lemma 4.8 to obtain Applying Theorem 3.2 we have Letting the theorem follows.

Bounding the core volume from above
Our goal in this section will be to prove the following theorem.
Theorem 5.1. Given S, there are constants K 3 and K 4 so that if Q(X, Y ) ∈ QF (S) is a quasi-Fuchsian manifold and P X and P Y are pants decompositions for which X ∈ V (P X ) and Y ∈ V (P Y ) then we have Given Theorem 4.1 and Theorem 3.2, Theorem 5.1 represents the final step in the proof of Theorem 1.2.
Let G ⊂ P(S) be a shortest path joining P X and P Y so that the length of G is simply d P (P X , P Y ). Let spin(G) = {α * | α ∈ P, P ∈ G} denote the geodesic representatives in Q(X, Y ) of elements of the pants decompositions along G. We call these geodesics the spinning geodesics for G; they will serve to anchor various tetrahedra in Q(X, Y ) at their vertices; we will then "spin" these tetrahedra by pulling their vertices around the geodesics.
Our upper bound for vol(X, Y ) will come from a model manifold N = S × I comprised of blocks that are adapted to spin(G), together with a piecewise C 1 surjective homotopy equivalence f : N → N ǫ (C(X, Y )) so that the image of each block under f has uniformly bounded volume. The model will decompose into two parts.
(1) The Caps: At each end of N are caps, namely products S × I on which f restricts to homotopies of simplicial hyperbolic surfaces realizing P X and P Y to the boundary components X ǫ h and Y ǫ h of the ǫneighborhood N ǫ (C(X, Y )) of the convex core.
(2) The Triangulated Part: The caps sit at either end of the triangulated part N ∆ , a union of tetrahedra on which f is simplicial: f lifts to a map sending each simplex to the convex hull of its vertices. It follows that the image of each tetrahedron ∆ ∈ N ∆ under f has uniformly bounded volume. We use the geodesics α * , where α ∈ P ∈ G, as a scaffolding to build N ∆ , a glueing of tetrahedra whose image interpolates between the simplicial hyperbolic surfaces Z X and Z Y . After "spinning" f sufficiently far about the spinning geodesics, all but a constant times d P (P X , P Y ) of the tetrahedra in N ∆ have images with small volume.
These two arguments give the desired bound after collecting constants.

Remark:
The above spinning trick is inspired by the ideal simplicial maps of [Th2] which are in effect a limit of the spinning process we perform here. The result in our context of passing to such a limit is an ideal triangulation of all but a bounded volume portion of C(X, Y ), with a uniformly bounded number of ideal tetrahedra necessary to accomplish each individual elementary move (the small volume tetrahedra collapse to lower dimensional ideal edges and faces). We have chosen to work with finite triangulations in the interest of demonstrating how the combinatorics of P(S) may be used to produce triangulations of 3-manifolds an semi-algorithmic manner, independent of any geometric structure.

Triangulations of surfaces.
We specify a type of triangulation of S that is suited to a pants decomposition P . By a pair of pants we will mean a connected component S of S − N (P ), the complement of the union of pairwise disjoint open annular neighborhoods N (P ) of the curves in P on S.

Definition 5.2. A standard triangulation T ( S) for a pair of pants S is a triangulation with the following properties:
(1) T ( S) has two vertices on each boundary component.
(2) T ( S) has two disjoint spanning triangles with no vertices in common, and a vertex on each component of ∂ S.
(3) The remaining 3 quadrilaterals are diagonally subdivided by an arc that travels "left to right" with respect to the inward pointing normal to ∂ S (see Figure 6).
We construct a standard triangulation suited to a pants decomposition P ∈ P(S) by gluing together standard triangulations on pairs of pants S as follows.
Definition 5.3. Given a pants decomposition P ∈ P(S), a standard triangulation suited to P is a triangulation T of S obtained as follows (see Figure 6): (1) T has two vertices p α andp α on each component α of P , and two edges e α andē α in the complement α − p α ∪p α .  Moves on triangulations. Given an elementary move on pants decompositions (P, P ′ ), i.e. d P (P, P ′ ) = 1, we now describe simple moves on triangulations that allow us to move from a standard triangulation suited to P to a standard triangulation suited to P ′ . To distinguish moves on triangulations from moves on pants decompositions, we refer to the latter as pants moves.
To fix notation, given a pants move (P, P ′ ) let α ∈ P and β ∈ P ′ be the curves for which i(α, β) = 0. We call α and β the curves involved in the pants move (P, P ′ ). Let N (P − α) denote the union of pairwise disjoint open annular neighborhoods about the curves in P −α, and let S α denote the essential subsurface of S − N (P − α) containing α. If S α has genus 1 then (P, P ′ ) is called a genus 1 pants move. Likewise, S α has genus 0 then (P, P ′ ) is called a genus 0 pants move. We say the pants move (P, P ′ ) occurs on S α .
A standard triangulation T suited to P naturally identifies candidate elementary moves for each α ∈ P : there is a natural choice of isotopy class of simple closed curves β ⊂ S α for which i(α, β) = 1 or i(α, β) = 2 depending on whether S α has genus 1 or genus 0. If S α has genus 1, then each spanning triangle for T in S α has one edge with its endpoints identified. These edges are in the same isotopy class which we call β(α, T ). Likewise, if S α has genus 0, then removing the edges of T that do not have endpoints lying on α produces two hexagons in the complement of the remaining edges. Concatenating edges in these hexagons joining the two vertices in each that lie on α we obtain an isotopy class of simple closed curves, which we again call β(α, T ).
There are three basic types of moves on these triangulations: MVI. The Dehn twist move. One standard move on triangulations we will use effects a Dehn twist of a standard triangulation suited to P about a curve α ∈ P . Given a triangulation T suited to P , let T W α (T ) denote the standard triangulation suited to P obtained by shifting each edge with a vertex on α to the right along α until it hits the next vertex. Then T W α (T ) is isotopic to the image of T under a right-α Dehn twist. We define T W −1 α (T ) similarly, by shifting edges to the left rather than to the right.
Note that for any triangulation T suited to P and any α ∈ P we have where τ α is a right α-Dehn twist. The other elementary moves on triangulations will be specific to a given type of pants move. Given a triangulation T suited to P and α ∈ P , we describe blowdown and blow-up moves that allow us to pass from a triangulation suited to P to a triangulation suited to the pants decomposition (P − α) ∪ β(α, T ). MVII. Genus 1 moves. Given a standard triangulation T suited to P the edges of T that close to form loops in the isotopy class of β(α, T ) bound an annulus on S. We call this annulus A α the α compressing annulus for T . The genus 1 α-blow down BD 1 α (T ) of T is the triangulation of S obtained by collapsing the arc e of α ∩ A α to a point and collapsing the two triangles in T containing e to a single edge.
MVIII. Genus 0 moves. Let T be a standard triangulation suited to P . Let e α andē α denote the two edges of T that constitute the curve α. Call these curves the α-edges of T . As described above there are two hexagons H andH obtained by removing all edges of T that do not have endpoints on α. The genus 0 α-blow-down BD 0 α (T ) = T ′ (Figure 9) is obtained by performing 3 "diagonal switches" on each hexagon (Figure 8) to yield a new triangulation with edges e β andē β as edges whose concatenation gives the curve β = β(α, T ).
Likewise, the genus 0 β-blow-up BU 1 β (T ′ ) is obtained by modifying the hexagons containing the α-edges by the inverses of the 3-diagonal switches. Note that BU 0 β • BD 0 α (T ) is a standard triangulation suited to the pants decomposition (P − α) ∪ β. We summarize properties of these moves as a lemma.
Lemma 5.4. Let (P, P ′ ) be a pants move involving α ∈ P and α ′ ∈ P ′ . Let T be a standard triangulation suited to P . Then there is an n ∈ Z so that BU α ′ • BD α • T W n α (T ) is a standard triangulation suited to P ′ . 5.2. Realizing moves by blocks. As before let Q(X, Y ) be a quasi-Fuchsian manifold, and let P X and P Y be pants decompositions for which X ∈ V (P X ) and Y ∈ V (P Y ). Let G ⊂ P(S) be a geodesic joining P X and P Y . Recall we denote by spin(G) = {α * | α ∈ P, P ∈ G} the spinning geodesics associated to G.
Equip each spinning geodesic α * with a pair of antipodal vertices p α andp α : i.e. points on α * so that the distance from p α top α along α * is maximal. For reference, we equip each α and thence each α * with an orientation.
Let P i , i = 0, . . . , m, denote the pants decompositions along the geodesic G, so that P 0 = P X and P m = P Y . Making an initial choice of standard triangulation T X suited to P X , Lemma 5.4 provides a sequence of moves on triangulations allowing us to process from the triangulation T X to a standard triangulation T Y suited to Y via standard triangulations T i suited to P i . We begin with a model manifold N 0 ∼ = S × I and triangulate ∂ + N 0 = S × {0} by T X . Our aim is to build models N i ∼ = S × I by gluing triangulated I-bundles to ∂ + N i−1 so that the resulting triangulation on 28 JEFFREY F. BROCK ∂ + N i = T i . We do this by building a triangulated subsurface block corresponding to each elementary move and successively gluing the blocks to ∂ + N i−1 .
Definition 5.5. Given a curve α in a pants decomposition P , we define the subsurface block by the quotient We denote the upper and lower boundary of B α by We now describe block triangulations associated to each elementary move on triangulations. We will say a triangulation T of a subsurface block B α realizes an elementary move T → M(T ) if we have Block triangulations. Let T be a standard triangulation on S α suited to α. Then the standard block triangulation T α is obtained from T × [0, 1]/ ∼ in the following way. Initially, T × [0, 1]/ ∼ is a cell decomposition of B α . For any edge e of T with ∂e ∩ ∂S α = ∅, e × [0, 1] is a quadrilateral to which we add a diagonal depending on the genus of S α .
• When S α has genus 0, the α-edges e α andē α determine quadrilaterals e α × [0, 1] andē α × [0, 1] in T . We triangulate these quadrilaterals with two new edges that run in the same direction along the annulus and cone off the new edges down to the vertices opposite the quadrilaterals (see Figure 10).ē • When S α has genus 1 in addition to the two edges e α andē α that concatenated give α there are 3 other edges e 1 , e 2 and e 3 that triangulate the α-compressing annulus A α for which ∂e j ∩ ∂S α = ∅, j = 1, 2, 3. We triangulate the annulus as before, and extend this to a triangulation of A α × [0, 1] with no new vertices as in Figure 10.
From the standard block triangulation, we build four types of blocks: BLI. The Dehn twist block. Given S α , the block triangulation T (T W α ) realizing the move T W α is obtained form the standard block triangulation T α as follows.
Consider the annulus A = α × [0, 1] in B α with the triangulation T A on A induced by T α . The reference orientation for α locally determines a left and right side of α in S α , and hence a left and right side of A in B α . Cut B α along A to obtain two annuli A L and A R that bound the local left and right side of B α − A. Re-glue the α × {0} boundary components of A L and A R by the identity, and re-glue the α × {1} boundary components of A L and A R shifted by a right Dehn twist. The triangulations induced by T A on A L and A R determine a triangulation of the torus A L ∪ A R after re-gluing. This triangulation naturally extends to a triangulation of a solid torus V with boundary A L ∪ A R by filling in tetrahedra (the triangulations on A L and A R differ by two pairs of diagonal switches. See Figure 11). After filling in by V , the result is a standard block B α with triangulation T tw+ α realizing the move T W α . We call the standard block B α equipped with the triangulation T tw+ BLII. Blow-up and blow-down blocks. We modify the standard block triangulation to obtain triangulated blocks that realize blow-up and blow-down moves as follows.
To realize a genus 1 α-blow-down by a block triangulation, we modify the standard block triangulation T α on B α by collapsing the compression annulus on ∂ + B α as in the description of the move BD 1 α . The only difference is that here in addition to collapsing triangles to edges we also collapse tetrahedra to triangles.
Similarly, if β = β(α, T ), we obtain a block triangulation realizing the genus 1 β-blow-up BU 1 β by collapsing the β compression annulus on ∂ − B β . The genus 0 α-blow-down (and blow-up) moves come from diagonal switches on hexagons. We realize the move BD 0 α by gluing tetrahedra realizing each of the diagonal switches to ∂ + B α . Likewise, the block triangulation realizing the genus 0 β-blow-up can be obtained by gluing tetrahedra in this way to the β-hexagons on We denote the α-blow-up block by B bu α and the α-blow-down block by B bd α . BLIII. Straightening blocks. There are two other types of block triangulations we will need that realize the identity move on a blown-down triangulation. We call these triangulated blocks straightening blocks. Given the triangulation BD 1 α (T ) or BD 0 α (T ) for a standard triangulation T suited to a pants decomposition P containing α, the genus g straightening triangulation, g = 0, 1 is obtained by completing the cell decomposition (BD g α (T ) ∩ S α ) × [0, 1]/ ∼ of B α , and extending this decomposition to a triangulation of B α with no new vertices. It is easy to check that this can be done. We denote the straightening blocks by B st α,β where β = β(α, T ). 5.3. Mapping in blocks and building the model. We now use our block triangulations to build a model manifold N ∼ = S × I. We build N in stages corresponding to standard triangulations T j suited to pants decompositions P j that intervene between P X and P Y respectively. At each stage there is a model N j also homeomorphic to S × I so that the top boundary component ∂ + N j = S × {1} is triangulated by the jth triangulation T j in the sequence of triangulations. Each N j will be obtained from N j−1 by attaching a triangulated block to ∂ + N j−1 that realizes the elementary move needed to move from T j−1 to T j .
The model N will come equipped with a map f : N → C(X, Y ) that is simplicial on each block: Definition 5.6. An incompressible mapping f : N → M from a triangulated 3manifold to a hyperbolic 3-manifold is simplicial if the lift f : N → H 3 sends each k-simplex to the convex hull of its vertices.
The following theorem describes the properties of our model and its mapping to the quasi-Fuchsian manifold Q(X, Y ). For simplicity we assume for the remainder of this section that S is closed, and detail the necessary modifications to the argument at the end.
Theorem 5.7. Given Q(X, Y ), there is a model manifold N ∼ = S × I, equipped with a surjective homotopy equivalence to the ǫ-neighborhood of the convex core C(X, Y ) with the following properties: (1) N is the union N = cap X ∪ N ∆ ∪ cap Y of the caps cap X ∼ = S × I and cap Y ∼ = S × I and the triangulated part N ∆ , a union of blocks of the above type glued top boundary to bottom boundary, all but 3d P (P X , P Y ) of which are Dehn twist blocks.
(2) The map f is piecewise C 1 and is simplicial on N ∆ .
(3) For each tetrahedron ∆ in N ∆ that lies in a Dehn twist block, f maps some edge of ∆ to a spinning geodesic. (4) The restriction f | cap X is a homotopy from ∂ − N ǫ (C(X, Y )) to a simplicial hyperbolic surface realizing P X , and the restriction f | cap Y is a homotopy from a simplicial hyperbolic surface realizing P Y to ∂ + N ǫ (C(X, Y )).
Proof: To motivate the construction of N we build the map f in stages as well.
Mapping in cap X . Let T X and T Y be standard triangulations suited to P X and P Y . Let determine a homotopy of the convex core boundary g X : Notice that this implies that h X realizes each curve α ∈ P X . Let N 0 = S × [0, 1], denote the domain for f X ; we will refer to N 0 = cap X as the X-cap of N . The top boundary component ∂ + N 0 carries the triangulation T X . Working inductively, we assume given a model N j at the jth stage: i.e.
(1) the model N j ∼ = S × I, consists of the X-cap and a triangulated part so that the upper boundary ∂ + N j is triangulated by a standard triangulation T j suited to P j , (2) N j comes equipped with a map f j : N j → C(X, Y ) that is simplicial on the triangulated part of N j , (3) f j | ∂ + Nj factors through a simplicial hyperbolic surface h j : Z j → C(X, Y ) with associated triangulation T j so that h j sends vertices v γ andv γ on γ ∈ P j to p γ andp γ on γ * .
Let α ∈ P j and α ′ ∈ P j+1 be the curves involved in the genus g elementary move (P j , P j+1 ), and let n be the integer guaranteed by Lemma 5.4 for which BU g α ′ • BD g α • T W n α (T j ) = T j+1 is a standard triangulation suited to P j+1 .
We now specify how to add triangulated blocks to N j and extend f j simplicially over each additional block to obtain the next stage of the model N j+1 .
Mapping in Dehn twist blocks. If n is positive, we attach n right Dehn twist blocks to the model and extend f j over them in sequence, while if n is negative we do likewise with left Dehn twist blocks.
We assume n is positive; the negative case is identical with left Dehn twist blocks replacing right Dehn twist blocks. We attach a right Dehn twist block B tw+ α to ∂ + N j so that the triangulation on ∂ − B tw+ α agrees with T j ∩ S α to obtain a new model N j,1 . We extend f j over B tw+ α to obtain a map f j,1 as follows. Recalling that B tw+ α has the form S α × [0, 1]/ ∼, we set f j (x, t) = f j (x, 0). We then straighten f j on α × {1} to its geodesic representative so that the vertices v andv on α × {1} to p α andp α , and finally we straighten f j,1 by a homotopy to make it simplicial on B tw+ α . We note that every tetrahedron in the Dehn twist block has an edge that maps to a geodesic arc of α * , verifying part (3).
The map f j,1 factors through a simplicial hyperbolic surface still realizing P j with associated triangulation T W α (T j ) which we denote by T j,1 . Repeating this procedure to add n Dehn twist blocks we arrive at a model N j,n equipped with a map f j,n : N j,n → C(X, Y ) so that (1) ∂ + N j,n carries the triangulation T j,n = T W n α (T j ), (2) f j,n factors through a simplicial hyperbolic surface realizing P j with associated triangulation T j,n , and (3) the vertices of T j,n map to p γ andp γ on γ * for each γ ∈ P j .
Mapping in blow-down blocks. Our discussion of how to attach a blow-down block to N j,n and how to extend f j,n over this block breaks into cases as usual.
Genus 0. The genus 0 blow-down block B bd α is attached to ∂ + N j,n along S α so that the triangulations agree as before. This gives a new model N j,bd . We extend f j,n over B bd α to give a map f j,bd : N j,bd → C(X, Y ) by first mapping B bd α to ∂ + N j,n , as with the Dehn twist block, and then straightening f j,bd to a simplicial map.
Genus 1. In the genus 1 case is the same, except that as there is only one vertex v on α in ∂ + B bd α , we simply send v to p α and straighten f j,bd to a simplicial map as before.
In each case the resulting map f j,bd | ∂ + N j,bd factors through a simplicial hyperbolic surface that realizes P j and has associated triangulation BD 0 α (T j,n ) which we denote by T j,bd .
Mapping in straightening blocks. Straightening blocks allow us to pass from simplicial hyperbolic surfaces realizing P j to simplicial hyperbolic surfaces realizing P j+1 . We attach the straightening block B st α,α ′ to N j,bd to obtain a model N j,st . We extend f j,bd over B st α,α ′ to a map f j,st by defining f j,st (x, t) = f j,st (x, 0), and then straightening f j,st on α ′ × {1} ⊂ ∂ + B st α,α ′ . We send the α ′ vertex to p α ′ or vertices to p α ′ andp α ′ and straighten the map to a simplicial map on B st α,α ′ . Now f j,st | ∂ + Nj,st factors through a simplicial hyperbolic surface realizing P j+1 with associated triangulation T j,bd once again.
Mapping in blow-up blocks. This procedure is essentially the inverse of the attaching and mapping in the blow-down blocks.
, and extend f j,st to f j,bu over B bu α ′ by setting f j,bu (x, t) = f j,bu (x, 0) and then straightening f j,bu rel-α ′ × {1} to a simplicial map.
The map f j,bu | ∂ + N j,bu factors through a simplicial hyperbolic surface realizing P ′ with associated triangulation T j+1 = BU α ′ • BD α • T W n α (T j ) which is a standard triangulation suited to P j+1 . This completes the inductive step.
Mapping in cap Y . Let |G| = d P (P X , P Y ) denote the length of a geodesic G ⊂ P(S). Then the above inductive procedure results finally in a map so that the restriction f |G|,0 | ∂ + N |G| factors through a simplicial hyperbolic surface realizing P Y with associated triangulation T Y (a standard triangulation suited to P Y ), and so that the vertices v γ andv γ map to the vertices p γ andp γ on the closed geodesic γ * for each γ ∈ P Y . We complete our model N by adding a Y -cap: this is a homotopy f Y : S × I → C(X, Y ) from f |G| | ∂ + N |G| to the convex core boundary g Y : Y h → ∂ + C(X, Y ). Gluing this homotopy S × I, to ∂ + N |G| and extending f |G| over the Y -cap by f Y , we obtain the final piece of our model N and the resulting map a homotopy equivalence whose restrictions Though the boundary ∂C(X, Y ) is not generically smooth, by taking the boundary ∂N ǫ (C(X, Y )) of the ǫ-neighborhood of the convex core we obtain a pair of C 1 surfaces ∂ + N ǫ (C(X, Y )) and ∂ − N ǫ (C(X, Y )) with C 1 path metrics [EM, Lem. 1.3.6]. In the interest of computing volume, we perturb f to a piecewise smooth map f ǫ : N → N ǫ (C(X, Y )) by adjusting f by a homotopy that changes f only on cap X and cap Y , so that are homeomorphisms, and f ǫ is C 1 on the interiors of the caps of N . The map f is already simplicial on the triangulated part of N so f ǫ is piecewise C 1 . A degree argument shows f ǫ is surjectve, proving the theorem. 5.4. Bounding the volume. Given a piecewise differentiable 3-chain C in a hyperbolic 3-manifold M , the function deg C : M → Z which measures the degree of C in M is well defined at almost every point of M . We define the mass mass(C) of C to be the integral where dV is the hyperbolic volume form on M (cf. [Th2,§4]).
Moreover, if F : P → M is a map of a piecewise differentiable 3-manifold P to M , and C is a piecewise differentiable 3-chain in P , then we define the F -mass of C by the integral The F -mass of C bounds the volume vol(F (C)) of the image of C in M . Hence, given our piecewise differentiable surjective map f ǫ : N → N ǫ (C(X, Y )) the volume vol(N ǫ (C(X, Y )), which bounds vol(X, Y ), is bounded by the sums of the f ǫ -masses of the chains that make up N . In other words, if N decomposes into 3-chains C k , we have Thus, Theorem 5.1 will follow from the following proposition.
Proposition 5.8. Given S there are constants K 3 > 1 and K 4 > 0 so that the map f ǫ is properly homotopic to a map f ǫ θ : N → N ǫ (C(X, Y )) for which Let V 3 denote the maximal volume of a tetrahedron in hyperbolic 3-space (see [Th1,ch. 7], [BP]). We begin our approach to Proposition 5.8 with the following lemma.
Lemma 5.9. There is a constant K ∆ so that the map f ǫ is properly homotopic to a map f ǫ θ that also satisfies the conclusions of Theorem 5.7 so that Proof: Because of the possibility of a large number of Dehn twist blocks in N ∆ , there is not in general a uniform constant K for which the number of tetrahedra used to triangulate N ∆ is less than Kd P (P X , P Y ). We will show, however, that by modifying f ǫ by a homotopy, we can force the tetrahedra that lie in Dehn twist blocks to have mass as small as we like. By Theorem 5.7, the number of blocks in N ∆ that are not Dehn twist blocks is bounded by 3d P (P X , P Y ). Since there is a uniform bound to the number of tetrahedra in any block, there is a constant K ∆ so that all but at most K ∆ d P (P X , P Y ) tetrahedra of N ∆ lie in Dehn twist blocks.
Let α * ∈ spin(G) be a spinning geodesic. Let f ǫ θ be defined by the following homotopy of f ǫ through maps that are simplicial on N ∆ : for each α ∈ ∪ j P j , slide the vertices p α andp α along the geodesic α * a distance in the direction of the reference orientation chosen for α. (See [Th2] for another example of this spinning of triangulations). The following lemma shows that tetrahedra that lie in a Dehn twist block, which we will call Dehn twist tetrahedra, can be made to have small f ǫ θ -mass by spinning to sufficiently high values of θ. Lemma 5.10. If ∆ ⊂ N ∆ is a Dehn twist tetrahedron, then mass f ǫ θ (∆) → 0 as θ → ∞.
Proof: Recall from Theorem 5.7, each Dehn twist tetrahedron ∆ has at least one edge e for which f ǫ (e) ⊂ α * for some spinning geodesic α * .
Lift f ǫ θ to f ǫ θ : N → H 3 and choose a lift ∆ to N . Let α * denote the lift of α * to H 3 for which f ǫ θ sends the lifted edge e ⊂ ∆ to α * . Let e ′ be the opposite edge of ∆ (e and e ′ have no endpoints in common).
Let I θ be the ideal tetrahedron in H 3 for which (1) f ǫ θ ( e ′ ) lies in one edge e ′ ∞ of I θ , (2) the two other edges e 1 ∞ and e 2 ∞ of I θ emanating from one endpoint of e ′ ∞ pass through endpoints p 1 and p 2 of f ǫ θ ( e). The image f ǫ θ ( ∆) lies in I θ (see Figure 12). Normalize by an isometry of H 3 so that the edge e ′ ∞ of I θ has its ideal endpoints at 0 and ∞, and so that the terminal fixed point of α * (with respect to the reference orientation for α) lies at 1 ∈ C.
Then for any r > 0 there exists a θ so that the two ideal vertices of I θ that do not lie at 0 and ∞ lie within a small disk |1 − z| < r. It follows that the dihedral angle of I θ at e ′ ∞ tends to 0 as θ → ∞. But the volume of an ideal tetrahedron tends to 0 as any of its dihedral angles tends to 0, so we have mass f ǫ θ (∆) < vol(I θ ) → 0 as θ → ∞.
Continuation of the proof of Lemma 5.9: By Lemma 5.10, for any Dehn twist block B, the quantity mass f ǫ θ (B) is as small as we like for θ sufficiently large. If B tw denotes the union of all Dehn twist blocks in N ∆ , then, we may choose θ sufficiently large so that mass f ǫ θ (B tw ) < 1. Since f ǫ θ is simplicial on N ∆ , we have mass f ǫ θ (∆) < V 3 for any tetrahedron ∆ ⊂ N ∆ , and since all but at most K ∆ d P (P X , P Y ) tetrahedra in N ∆ lie in B tw , we have mass f ǫ θ (N ∆ ) < K ∆ · V 3 · d P (P X , P Y ) + 1.
Bounding the volume of the caps. The bound on the f ǫ θ -mass of the triangulated part N ∆ in terms of the distance d P (P X , P Y ) will be sufficient for Proposition 5.8 once we show the following uniform bound on the f ǫ θ -mass of the caps of N .
Lemma 5.11. There is a uniform constant K cap , depending only on S so that after modifying f ǫ θ by a homotopy on cap X we have mass f ǫ θ (cap X ) < K cap and similarly for cap Y .
Proof: Our goal will be to modify the homotopy f ǫ θ | cap X from ∂ − N ǫ (C(X, Y )) to the simplicial hyperbolic surface Z X by cutting the surface S into annuli and controlling the trace of the homotopy on each annulus (a solid torus). To obtain such control, we choose this decomposition compatibly with the pants decomposition P X .
We fix attention on a single pair of pants S ⊂ S − P X . By a figure-8 curve on S we will mean a closed curve that intersects itself once on S and divides S into three annuli, one parallel to each boundary component of S (see Figure 13). To prove Lemma 5.11 we establish the following basic lemma in hyperbolic surface geometry. (We continue to treat the closed case; the case when S has boundary is similar).
Lemma 5.12. Let S a closed surface with negative Euler characteristic, and let L ′ ≥ L be a constant greater than or equal to the Bers constant L for S. Then there is a constant L 8 (L ′ ) > 0 so that the following holds: if P is a pants decomposition of S, Z ∈ Sing k (S) is a possibly singular hyperbolic surface, and the length bound ℓ Z (α) < L ′ holds for each α ∈ P , then any figure-8 curve γ in any component S ⊂ S − P satisfies ℓ Z (γ) < L 8 (L ′ ). Proof: Note that any bound on the geodesic length of a given figure-8 curve γ guarantees a bound on the geodesic length of any other, by taking twice the original bound. Assume first that Z ∈ Teich(S), so Z is a non-singular hyperbolic surface. From the thick-thin decomposition for hyperbolic surfaces, given any δ > 0, there is a uniform bound D δ to the diameter diam(Z ≥δ ) of the δ-thick part of Z, where D δ depends only on δ and the surface S. Let ∂ S = α 1 ⊔ α 2 ⊔ α 3 . Since ℓ Z (α i ) < L ′ , there is a δ so that α * i can only intersect Z <δ in a component of Z <δ for which it is the core curve. Choosing δ smaller if necessary, we can ensure that the boundary of any component of Z <δ is a curve of length less than L ′ .
Let Z be the realization of S as a subsurface of Z bounded by the curves ∪α * i . Either two of the α * i lie entirely in Z ≥δ or two of the α i are homotopic into Z <δ . Without loss of generality, assume α * 1 and α * 2 lie in Z ≥δ . Then they can be joined by an arc b in Z of length less than D δ . Either α 1 · b · α 2 · b −1 or α 1 · b · α −1 2 · b −1 is a figure-8 curve of length less than 2L ′ + 2D δ . Otherwise, if A 1 and A 2 are components of Z <δ representing the homotopy classes of α 1 and α 2 , then there are components a 1 of ∂A 1 and a 2 of ∂A 2 with length less than L ′ , and an arc b joining a 1 to a 2 in Z of length less than D δ . By the same reasoning, there is a figure-8 curve γ of length To treat the potentially singular case, let Z ∈ Sing k (S) satisfy ℓ Z (α i ) < L ′ for each α i ∈ P . Let Z h ∈ Teich(S) represent the hyperbolic surface in the same conformal class as Z. If each α i ∈ P satisfies ℓ Z h (α i ) < L ′ then there is a figure-8 curve γ ⊂ S with ℓ Z h (γ) < 2L ′ +2D δ by the above reasoning. By Ahlfors' lemma, [Ah] we have ℓ Z (γ) < ℓ Z h (γ) < 2L ′ + 2D δ proving the lemma in this case.
If, however, some α i ⊂ ∂ S satisfies ℓ Z h (α i ) ≥ L ′ ≥ L, then by Theorem 2.1 there is a simple closed curve β, with i(α i , β) = 0, for which ℓ Z h (β) < L. Again applying Ahlfors' lemma, the curve β has length ℓ Z (β) < L on Z, so its geodesic representative β * furnishes an arc b in S that joins α * i to α * j , or joins α * i to itself. In either case, two copies of b together with arcs in the geodesics at its endpoints can be assembled to form a figure-8 curve β ⊂ S with length bounded by ℓ Z (β) < 2L ′ + 2L ≤ 4L ′ .
Remark: Though we continue to work in the setting where S is closed, the proof